Combinatorics of free and simplicial line arrangements

Abstract

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.

18 pages, 2 figures. Updates in this version: the exposition has been changed in order to increase the level of organization of the paper (results are unchanged). The focus has been shifted more towards pseudoline arrangements, which is appropriate because most results are combinatorial in nature